Self Organization in a Diffusion Model of Thin Electric

Self Organization in a Diffusion Model of Thin Electric Current Sheets Student Researcher: Andrew Kercher Faculty Advisor: Dr. Robert Weigel Undergraduate Research in Computational Mathematics George Mason University April 5, 2008 Andrew Kercher (GMU) SOC April 5, 2008 1 / 14 Introduction Upcoming Topics Space Weather Solar Wind Earth’s Magnetosphere Andrew Kercher (GMU) SOC April 5, 2008 2 / 14 Introduction Upcoming Topics Space Weather Solar Wind Earth’s Magnetosphere Self Organized Criticality Instability Andrew Kercher (GMU) SOC April 5, 2008 2 / 14 Introduction Upcoming Topics Space Weather Solar Wind Earth’s Magnetosphere Self Organized Criticality Instability Prior Work of Lu and Klimas Andrew Kercher (GMU) SOC April 5, 2008 2 / 14 Introduction Upcoming Topics Space Weather Solar Wind Earth’s Magnetosphere Self Organized Criticality Instability Prior Work of Lu and Klimas Analysis Model Total Field Energy Time Averaged Mean Power Spectral Density Andrew Kercher (GMU) SOC April 5, 2008 2 / 14 Introduction Upcoming Topics Space Weather Solar Wind Earth’s Magnetosphere Self Organized Criticality Instability Prior Work of Lu and Klimas Analysis Model Total Field Energy Time Averaged Mean Power Spectral Density Conclusions Andrew Kercher (GMU) SOC April 5, 2008 2 / 14 Introduction Upcoming Topics Space Weather Solar Wind Earth’s Magnetosphere Self Organized Criticality Instability Prior Work of Lu and Klimas Analysis Model Total Field Energy Time Averaged Mean Power Spectral Density Conclusions Future Work Andrew Kercher (GMU) SOC April 5, 2008 2 / 14 Space Weather Solar Wind Space Plasma Andrew Kercher (GMU) SOC April 5, 2008 3 / 14 Space Weather Solar Wind Space Plasma Magnetosphere Region in space, surrounding the earth, composed of charged particles and governed by magnetic flux. Andrew Kercher (GMU) SOC April 5, 2008 3 / 14 Space Weather Solar Wind Space Plasma Magnetosphere Region in space, surrounding the earth, composed of charged particles and governed by magnetic flux. Plasma Sheet Site of Reconnection Andrew Kercher (GMU) SOC April 5, 2008 3 / 14 Goals Describe the physical interactions within the system: Lead to better predictions when forecasting space weather Aid Development of a Physical Theory for this system Andrew Kercher (GMU) SOC April 5, 2008 4 / 14 Self Organized Criticality Simply rules govern the dynamics Thresholds exist within the system The threshold is eventually exceeded by the build up of energy Systems displaying characteristics associated with SOC dissipate stored energy in avalanches [3]. Andrew Kercher (GMU) SOC April 5, 2008 5 / 14 Instability ∂ B(z, t) ∂t ∂ = Dmin ∂z2 B(z, t) + S(z) 2 ∂ B(z, t) ∂t = ∂ ∂z “ ” ∂ D(z, t) ∂z B(z, t) + S(z) SOC April 5, 2008 6 / 14 Andrew Kercher (GMU) Instability ∂ B(z, t) ∂t ∂ = Dmin ∂z2 B(z, t) + S(z) 2 ∂ B(z, t) ∂t = ∂ ∂z “ ” ∂ D(z, t) ∂z B(z, t) + S(z) SOC April 5, 2008 6 / 14 Andrew Kercher (GMU) Model Description The Model is Given by: ∂Bx ∂ = ∂t ∂z D(z, t) ∂Bx ∂z + S(z) (1) Q ∂Bx ∂ ∂z (D(z, t)) = ∂t τ Q ∂Bx ∂z = − D τ (2) Dmin for low state Dmax for high state (3) Andrew Kercher (GMU) SOC April 5, 2008 7 / 14 Time 1 Time 4 The input of energy by the source term drives the system to the point of criticality Time 2 Time 5 Time 3 Andrew Kercher (GMU) Time 6 SOC April 5, 2008 8 / 14 Time 1 Time 4 The input of energy by the source term drives the system to the point of criticality Once the critical point is reached, the system reacts by unloading the energy in avalanches Time 2 Time 5 Time 3 Andrew Kercher (GMU) Time 6 SOC April 5, 2008 8 / 14 Time 1 Time 4 The input of energy by the source term drives the system to the point of criticality Once the critical point is reached, the system reacts by unloading the energy in avalanches Time 2 Time 5 System returns to a stable state, but steep slopes will be present in many local spatial positions [3]. Time 3 Andrew Kercher (GMU) Time 6 SOC April 5, 2008 8 / 14 Total Energy The total energy of the system at any instant is defined: E(t) = (Bx )2 dz Figure 1: S0 = 3 × 10−4 Figure 3: S0 = 3 × 10−3 Figure 2: S0 = 10−4 Andrew Kercher (GMU) SOC Figure 4: S0 = 10−3 April 5, 2008 9 / 14 Time Averaged Mean Field strength balanced by induced dynamic state System remains close to, but under the critical state Andrew Kercher (GMU) SOC April 5, 2008 10 / 14 Parameters Free Parameters: τ , Dmin , Dmax , S0 , k, β Andrew Kercher (GMU) SOC April 5, 2008 11 / 14 Parameters Free Parameters: τ , Dmin , Dmax , S0 , k, β Reduction in the number of parameters To analyze the effects of the free parameters on the systems ability to attain an SOC state Dr , S0 , β Dmin Where Dr is the ratio of Dmin and Dmax , i.e Dr = Dmax Andrew Kercher (GMU) SOC April 5, 2008 11 / 14 Parameters Free Parameters: τ , Dmin , Dmax , S0 , k, β Reduction in the number of parameters To analyze the effects of the free parameters on the systems ability to attain an SOC state Dr , S0 , β Dmin Where Dr is the ratio of Dmin and Dmax , i.e Dr = Dmax The system becomes: ∂Bx ∂ = ∂t ∂z ∂ ∂t Q Andrew Kercher (GMU) D(z , t ) ∂Bx ∂z + S0 sin ∂Bx ∂z πz 2L (4) D (z , t ) = Q ∂Bx ∂z = SOC −D (5) (6) April 5, 2008 11 / 14 Dr low state 1 high state Parameters Free Parameters: τ , Dmin , Dmax , S0 , k, β Reduction in the number of parameters To analyze the effects of the free parameters on the systems ability to attain an SOC state Dr , S0 , β Dmin Where Dr is the ratio of Dmin and Dmax , i.e Dr = Dmax The system becomes: ∂Bx ∂ = ∂t ∂z ∂ ∂t Q Andrew Kercher (GMU) D(z , t ) ∂Bx ∂z + S0 sin ∂Bx ∂z πz 2L (4) D (z , t ) = Q ∂Bx ∂z = SOC −D (5) (6) April 5, 2008 11 / 14 Dr low state 1 high state Analysis: Nondimensional Model Values of remaining parameters: Andrew Kercher (GMU) SOC April 5, 2008 12 / 14 Analysis: Nondimensional Model Values of remaining parameters: Dr = fixed, β = fixed, S0 = varied Movie (click to play) Andrew Kercher (GMU) SOC April 5, 2008 12 / 14 Conclusions and Future Work The Diffusive System Displays Characteristics associated with Self-Organized Criticality The number of states the system takes on is much greater then what was previously known. Need to research/develop a smarter algorithm for analysis. Andrew Kercher (GMU) SOC April 5, 2008 13 / 14 Thanks Special thanks to... Dr. Robert Weigel URCM Lu and Klimas, for their prior work and of course, the audience Andrew Kercher (GMU) SOC April 5, 2008 14 / 14 Per Bak, Chao Tang, Kurt Wiesenfeld, Phys. Rev. Lett. Vol. 59 (1987) 381 Henrik Heldtoft Jensen, Kim Christensen and Hans C. Fogedby, Phys. Rev.B, Vol. 40 (1989) 7425 A. Klimas et al., Self-organized substorm phenomenon and its relation to localized reconnection in the magnetospheric plasma sheet, J. Geophys. Res., 105(A8), (2000) 18,765-18,780. E. T. Lu, Avalanches in continuum driven dissipative systems, Phys. Rev. Lett., 74(13), (1995) 2511-2514. Andrew Kercher (GMU) SOC April 5, 2008 14 / 14